Minimal Faithful Representations of Split Extensions by Abelian Groups

Abstract

Every finite group admits a representation which is 'most efficient' in the sense that the representation is faithful and the dimension is minimal. We call such representations minimal faithful representations. The image of this representation retains all the information about the abstract group and can come with some additional geometric structure. For example, a minimal faithful representation could preserve a symplectic form or act on a fixed basis by permutations. It is the purpose of this paper to address these ideas for finite semi-direct products of groups split by an abelian group whose splitting homomorphism is faithful. We calculate the minimal faithful dimensions of such representations over both the complex and real numbers, and we express them in terms of group invariants to argue that in some sense these groups are extensions of tori. In the real case, the dimension of a minimal faithful representation and the structure it preserves encode the possible types of orbifold singularities of the group in question. We apply our framework to some families of groups to calculate their minimal faithful dimensions and analyze some of their orbifold singularities.